## Electronic Communications in Probability

### Local explosion in self-similar growth-fragmentation processes

#### Abstract

Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in [3], where special attention was given to the self-similar case. A Malthusian condition was notably given under which the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. Our main result in this work states the converse: when this condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes.

#### Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 66, 12 pp.

Dates
Accepted: 1 August 2016
First available in Project Euclid: 14 September 2016

https://projecteuclid.org/euclid.ecp/1473854582

Digital Object Identifier
doi:10.1214/16-ECP13

Mathematical Reviews number (MathSciNet)
MR3548778

Zentralblatt MATH identifier
1346.60038

#### Citation

Bertoin, Jean; Stephenson, Robin. Local explosion in self-similar growth-fragmentation processes. Electron. Commun. Probab. 21 (2016), paper no. 66, 12 pp. doi:10.1214/16-ECP13. https://projecteuclid.org/euclid.ecp/1473854582

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